258 . CENTRAL FOKCES. 



of the body moving in AVC, D6 or DF will also represent 

 the velocity at I of the body moving in V I K. Then take 

 D F = y as the centripetal force in D or I (that is, as any 

 power of the distance D C, or a x, V C being , and C D, x) 

 V D F L will be / y d z. Describe the circle V X Y with C V 



7 1 fl Z 



as radius. Let VX = z, and YX will be dz, and N K = . 



a 



Then ICK being as the time, and dt being constant, that 



I C x K X . 



triangle, or , is constant, and K .N is as a constant 



quantity divided by I C, or as . If we take to *J A V L B 



(proportioned to the force at any one point V and therefore 

 given), as KN to IK, therefore this will in all points be the 

 proportion ; and the squares will be proportional, or J* y d x : 



O a 



~ : : I K 2 , or K N 2 + I N 2 , to K N 2 ; and therefore fydx 



X 



Q 2 Q 2 x s d z* x d z 



: - - :: I N 2 , or d x* : . Therefore = 



x x a a 



Qd x a^a z 



; and multiplying by x, - - (twice the 



x* 



Qdx . x*dz 



sector I C K) = . Again a dz : : : a* : x* ; 



Q 2 



or 



J _ ~2 ~ 



tff \Jfif I* It- V^, \Ai tAS , - 



and adz = - - x - = ^ X = twice the 



a a* a* / Q 2 



x" 

 sector Y C X. 



Hence results this construction. Describe the curve a & Z, 



Q 



such that (D b = u) its equation shall be u = 



2 V fy dx ~* 



and the curve acx such that (D c = 0) its equation may be 



